One factor models of the term structure of interest rates were developed to provide insights into the valuation of fixed income options. Since the models’ early development from 1977 to 1993, they have been consistently misapplied in asset and liability (interest rate risk) management. Many analysts, choosing one of the common one factor models, create random interest rate scenarios to evaluate the interest rate risk of the firm. This blog post uses 50 years of U.S. Treasury yield curve information to prove that the major implications of one factor term structure models are inconsistent with historical data. A multi-factor approach is necessary to create realistic interest rate simulations. The author wishes to thank Professor Robert A. Jarrow for helpful comments. Any errors that remain are those of the author alone.

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In this blog post, we take the same approach as Jarrow, van Deventer and Wang (2003) in their paper “A Robust Test of Merton’s Structural Model for Credit Risk.” In that paper, the authors explained how, for any choice of model parameters, the Merton model of risky debt implies that stock prices would rise and credit spreads would fall if the assets of the firm were to rise in value. The opposite movements would occur if the assets of the firm were to fall in value. Jarrow, van Deventer, and Wang used a wide array of bond data and proved that this implication of the Merton model was strongly rejected by the data at extremely high levels of statistical significance.

We now take a similar approach to one factor term structure models. Speaking broadly, the original Macaulay (1938) duration concept can be thought of as the first one factor term structure model. Robert Merton (1970) developed a similar term structure model in a continuous time framework. Oldrich Vasicek (1977) brought a greater realism to one factor term structure modeling by introducing the concept of mean reversion to create interest rate cycles. Cox, Ingersoll, and Ross (1985) pioneered the affine term structure model, where the magnitude of random shocks to the short term interest rate varies with the level of interest rates. Ho and Lee (1986) and Hull and White (1993) developed one factor term structure models which could be fit exactly to observable yields. Black, Derman and Toy (1990) and Black and Karasinski (1991) focused on term structure models based on the log of the short rate of interest, insuring that rates would not be negative. Finally, Heath, Jarrow and Morton (1992) derived a general framework for N-factor term structure models that specifies the no arbitrage conditions that must apply to forward rate movements given the current level of the yield curve. For two excellent summaries of the literature on term structure models, please see Duffie and Kan (1994) and Jarrow (2009).

Among one factor term structure models, for any set of parameters, there is almost always one characteristic in common. The majority of one factor models make all interest rates positively correlated (which implies the time-dependent interest rate volatility sigma(T) is greater than 0). We impose this condition on a one factor model. The non-random time-dependent drift(T) in rates is small and, if risk premia are consistent and positive (for bonds) across all maturities T, then the drift will be positive as in the Heath Jarrow and Morton (1992) drift condition. With more than 200 business days per year, the daily drift is almost always between plus one and minus one basis point. Hence, one might want to exclude from consideration yield curve changes that are small, so the effect of the drift is removed. We ignore the impact of drift in what follows.

Given these restrictions, for a random shift upward in the single factor (usually the short rate of interest), the zero coupon yields at all maturities will rise. For a random shift downward in the single factor, all zero coupon yields will fall. If the short rate is unchanged, there will be no random shock to any of the other zero coupon yields either. Please see van Deventer, Imai and Mesler (2004) for the relevant equations for the Vasicek and Hull and White models.

Following Jarrow, van Deventer and Wang, we ask this simple question: “What percent of the time is it true that all interest rates either rise together, fall together, or stay unchanged?” If this implication of one factor term structure models is consistent with actual yield curve movements, we cannot reject their use. If this implication of one factor term structure models is inconsistent with actual yield movements, we reject their use in interest rate risk management and asset and liability management.

**Results from the U.S. Treasury Market, 1962-2011**

In this blog we use the same data as Dickler, Jarrow and van Deventer (2011): U.S. Treasury yields reported by the Federal Reserve in its H15 statistical release from January 2, 1962 to August 22, 2011. During this period, the Federal Reserve changed the maturities on which it reported data frequently:

We calculate the daily changes in yields over this period, eliminating the first daily change after a change in data regime, leaving 12,386 data points for analysis. We calculate daily yield changes on three bases:

- Using the yields from the Federal Reserve with no other analysis
- Using monthly forward rates
- Using monthly zero coupon bond yields, expressed on a continuous compounding basis

Monthly forward rates and zero coupon yields were calculated using the modified Adams and van Deventer (1994) approach employed by Dickler, Jarrow and van Deventer. We then asked the question posed above: “What percentage of the time were yield curve movements consistent with the implications of one factor term structure models?” Yield curve movements were considered consistent with one factor term structure models in three cases:

- Negative shift, in which interest rates at all maturities declined
- Positive shift, in which interest rates at all maturities increased
- Zero shift, in which interest rates at all maturities remained unchanged

By definition, any other yield curve movement consists of a combination of positive, negative and zero shifts at various maturities on that day. We label this a yield curve “twist,” something that cannot be explained by one factor term structure models.

The results are striking. Using only the input data from the H15 statistical release, with no yield curve smoothing, 7,716 of the 12,386 days of data showed yield curve twists, 62.3% of all observations. This is a stunningly high level of inconsistency with the assumptions of one factor term structure models.

For monthly zero coupon bond yields, yield curve twists occurred on 75.2% of the 12,386 days in the sample. When forward rates were examined, yield curve twists prevailed on 94.3% of the days in the sample. One factor term structure models would have produced results that are consistent with actual forward rates only 5.7% of the 1962-2011 period.

Using the approach of Jarrow, van Deventer and Wang (2003), we can now pose this question: is the consistency ratio we have derived different to a statistically significant degree from the 100% ratio that would prevail if the hypothesis of a one factor term structure is true?

We use the fact that the measured consistency ratio p is a binomial variable (true or not true, consistent or not consistent) with a standard deviation s as follows:

where p is the measured consistency ratio and n is the sample size, 12,386 observations. The chart below shows the standard deviations for each of the three data classes and the number of standard deviations between the measured consistency ratio p and the ratio (100%) that would prevail if the one factor term structure model hypothesis were true.

We reject the hypothesis that the consistency ratio is 100%, the level that would prevail if one factor term structure models were an accurate description of yield curve movements. The hypothesis that one factor term structure models are “true” is rejected with an extremely high degree of statistical significance.

**Implications of Results**

There are a number of very serious errors that can result from an interest rate risk and asset and liability management process that relies solely on the assumption that one factor term structure models are an accurate description of potential yield movements:

- Measured interest rate risk will be incorrect, and the degree of the error will not be known. Using the data above, 62.3% of the time actual yield curves will show a twist, but the modeled yield curves will never show a twist.
- Hedging using the duration or one factor term structure model approach assumes that interest rate risk of one position (or portfolio) can be completely eliminated with the proper short position in an instrument with a different maturity. The duration/one factor term structure model approach assumes that if interest rates on the first position rise, interest rates will rise on the second position as well so “going short” is the right hedging direction. The data above shows that on 62.3% of the days from 1962 to 2011, this “same direction” assumption was potentially false (some maturities will show same direction changes and some will show opposite direction changes) and the hedge could actually ADD to risk, not reduce risk.
- All estimates of prepayments and interest rate-driven defaults will be measured inaccurately
- Economic capital will be measured inaccurately
- Liquidity risk will be measured inadequately
- Non-maturity deposit levels will be projected inaccurately

This is an extremely serious list of deficiencies. The only remedy is to move as soon as possible to a more general N-factor model of interest rate movements. This should be done using the best available econometric results like those from Kamakura Risk Information Services and a simulation system like Kamakura Risk Manager. Academic assumptions about the stochastic processes have been too simple to be realistic. Jarrow (2009) notes,

“Which forward rate curve evolutions (HJM volatility specifications) fit markets best? The literature, for analytic convenience, has favored the affine class but with disappointing results. More general evolutions, but with more complex computational demands, need to be studied. How many factors are needed in the term structure evolution? One or two factors are commonly used, but the evidence suggests three or four are needed to accurately price exotic interest rate derivatives.”

This view is shared by the Basel Committee on Banking Supervision. In its December 31, 2010 *Revisions to the Basel II Market Risk Framework*, the Committee states its requirements clearly on page 12:

“For material exposures to interest rate movements in the major currencies and markets, banks must model the yield curve using a minimum of six risk factors.”

We conclude with a brief description of the interest rate data used.

**Background Information on U.S. Treasury Yield Curve Movements**

The website www.kamakuraco.com displays three videos of 50 years of daily movements of U.S. Treasury par coupon bond yields, U.S. Treasury monthly forward rates, and U.S. Treasury monthly zero coupon bond yields. The data is described pictorially in Dickler, Jarrow, and van Deventer (three papers dated 2011) and statistically in Dickler and van Deventer (three more papers dated 2011). Par coupon bond, zero coupon bond, and forward rate data is available by subscription from Kamakura Risk Information Services at info@kamakuraco.com.

**References**

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Black, Fischer, “Interest Rates as Options,” *Journal of Finance* (December), pp. 1371-1377, 1995.

Black, Fischer, E. Derman, W. Toy, “A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options,” *Financial Analysts Journal*, pp. 33-39, 1990.

Black, Fischer and Piotr Karasinski, “Bond and Option Pricing when Short Rates are Lognormal,” *Financial Analysts Journal*, pp. 52-59, 1991.

Cox, John C., Jonathan E. Ingersoll, Jr. and Stephen A. Ross, “An Analysis of Variable Rate Loan Contracts,” *Journal of Finance*, pp. 389-403, 1980.

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Dickler, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Forward Rates,” Kamakura memorandum, September 13, 2011.

Dickler, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields, Volume II: A Pictorial History of 50 Years of U.S. Treasury Zero Coupon Bond Yields,” Kamakura memorandum, September 26, 2011.

Dickler, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields, Volume III: A Pictorial History of 50 Years of U.S. Treasury Par Coupon Bond Yields,” Kamakura memorandum, October 5, 2011.

Dickler, Daniel T. and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An Analysis of 50 Years of Daily U.S. Treasury Forward Rates,” Kamakura blog, www.kamakuraco.com, September 14, 2011.

Dickler, Daniel T. and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An Analysis of 50 Years of Daily U.S. Treasury Zero Coupon Bond Yields,” Kamakura blog, www.kamakuraco.com, September 26, 2011.

Dickler, Daniel T. and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An Analysis of 50 Years of Daily U.S. Par Coupon Bond Yields,” Kamakura blog, www.kamakuraco.com, October 6, 2011.

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Jarrow, Robert A., Donald R. van Deventer, and Xiaoming Wang, “A Robust Test of Merton’s Structural Model for Credit Risk,” *Journal of Risk*, 6 (1), 2003.

Macaulay, Frederick R. *Some Theoretical Problems Suggested by Movements of Interest Rates, Bond Yields, and Stock Prices in the United States since 1856*. New York, Columbia University Press,1938.

Merton, Robert C. “A Dynamic General Equilibrium Model of the Asset Market and Its Application to the Pricing of the Capital Structure of the Firm,” Working Paper No. 497-70, A. P. Sloan School of Management, Massachusetts Institute of Technology, 1970. Reproduced as Chapter 11 in Robert C. Merton, Continuous Time Finance, Blackwell Publishers, Cambridge, Massachusetts, 1993.

van Deventer, Donald R. and Kenji Imai, *Financial Risk Analytics: A Term Structure Model Approach for Banking, Insurance, and Investment Management*, Irwin Professional Publishing, Chicago, 1997.

van Deventer, Donald R., Kenji Imai, and Mark Mesler, *Advanced Financial Risk Management*, John Wiley & Sons, 2004. Translated into modern Chinese and published by China Renmin University Press, Beijing, 2007.

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Donald R. van Deventer

Kamakura Corporation

Honolulu, Hawaii

November 7, 2011

© Copyright 2011 by Donald R. van Deventer, All Rights Reserved.